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The Poincaré lemma states that every closed differential form is locally exact.
Theorem (Poincaré Lemma) [1] Suppose $X$ is a smooth manifold, $\Omega^k(X)$ is the set of smooth differential $k$ -forms on $X$ , and suppose $\omega$ is a closed form in $\Omega^k(X)$ for some $k>0$ .
- Then for every $x\in X$ there is a neighbourhood $U\subset X$ , and a $(k-1)$ -form $\eta\in \Omega^{k-1}(U)$ , such that $$ d\eta = \iota^\ast \omega,$$ where $\iota$ is the inclusion $\iota:U\hookrightarrow X$ .
- If $X$ is contractible, this $\eta$ exists globally; there exists a $(k-1)$ -form $\eta\in \Omega^{k-1}(X)$ such that $$ d\eta = \omega.$$
Despite the name, the Poincaré lemma is an extremely important result. For instance, in algebraic topology, the definition of the $k$ th de Rham cohomology group $$ H^k(X) = \frac{ \operatorname{Ker}\{ d\colon \Omega^k(X)\to \Omega^{k+1}(X)\}}{ \operatorname{Im}\{ d\colon \Omega^{k-1}(X)\to \Omega^{k}(X)\}} $$ can be seen as a measure of the degree in which the Poincaré lemma fails. If $H^k(X)=0$ , then every $k$ form is exact, but if $H^k(X)$ is non-zero, then $X$ has a non-trivial topology (or ``holes'') such that $k$ -forms are not globally exact. For instance, in $X=\sR^2\setminus\{0\}$ with polar coordinates $(r,\phi)$ , the $1$ -form $\omega=d\phi$ is not globally exact.
- 1
- L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
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Cross-references: polar coordinates, de Rham cohomology group, topology, algebraic, contractible, inclusion, neighbourhood, smooth, smooth manifold, closed differential form
There are 4 references to this entry.
This is version 9 of Poincaré lemma, born on 2004-01-11, modified 2007-02-28.
Object id is 5509, canonical name is PoincareLemma.
Accessed 5699 times total.
Classification:
| AMS MSC: | 53-00 (Differential geometry :: General reference works ) |
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Pending Errata and Addenda
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